what is the shortest wavelength present in the Paschen series of spectral lines?

The Paschen series is a set of spectral lines that correspond to electronic transitions in hydrogen atoms, specifically when electrons fall from higher energy levels down to the third energy level (n=3). To determine the shortest wavelength in this series, we need to consider the transition that involves the highest energy level, which is when an electron drops from infinity (n=∞) to n=3.

Understanding the Calculation

The wavelength of the emitted light during these transitions can be calculated using the Rydberg formula:

1/λ = R_H (1/n1² - 1/n2²)

In this formula:

• λ is the wavelength of the emitted light.
• R_H is the Rydberg constant, approximately 1.097 x 10^7 m^-1.
• n1 is the lower energy level (for the Paschen series, n1 = 3).
• n2 is the higher energy level (for the shortest wavelength, n2 = ∞).

Plugging in the Values

For the shortest wavelength in the Paschen series:

n1 = 3 and n2 = ∞. Thus, the equation becomes:

1/λ = R_H (1/3² - 1/∞²)

Since 1/∞² is effectively 0, the equation simplifies to:

1/λ = R_H (1/9)

Now substituting the value of R_H:

1/λ = (1.097 x 10^7 m^-1) * (1/9)

1/λ = 1.219 x 10^6 m^-1

Finding the Wavelength

To find λ, we take the reciprocal:

λ = 1 / (1.219 x 10^6 m^-1)

Calculating this gives:

λ ≈ 8.2 x 10^-7 m

Converting this to nanometers (1 m = 10^9 nm):

λ ≈ 820 nm

Significance of the Result

The shortest wavelength in the Paschen series is approximately 820 nanometers, which falls within the infrared region of the electromagnetic spectrum. This means that while these transitions are not visible to the human eye, they play a crucial role in various applications, such as spectroscopy and understanding the behavior of hydrogen in different environments.

In summary, the shortest wavelength in the Paschen series occurs during the transition from n=∞ to n=3, resulting in a wavelength of about 820 nm. This calculation illustrates the relationship between energy levels in atoms and the wavelengths of light they emit, which is fundamental to the study of atomic physics and quantum mechanics.